↑

Properties

Label 8-72e4-1.1-c8e4-0-0
Degree $8$
Conductor $26873856$
Sign $1$
Analytic cond. $740155.$
Root an. cond. $5.41583$
Motivic weight $8$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 816·7-s − 2.27e4·13-s − 7.60e4·19-s + 2.89e5·25-s − 4.05e6·31-s − 6.35e6·37-s − 1.86e7·43-s − 2.01e7·49-s − 1.17e7·61-s − 2.64e7·67-s + 1.09e8·73-s − 4.18e7·79-s + 1.85e7·91-s + 4.28e8·97-s + 7.49e7·103-s + 4.60e8·109-s + 4.48e8·121-s + 127-s + 131-s + 6.20e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.339·7-s − 0.797·13-s − 0.583·19-s + 0.740·25-s − 4.38·31-s − 3.39·37-s − 5.46·43-s − 3.49·49-s − 0.848·61-s − 1.31·67-s + 3.84·73-s − 1.07·79-s + 0.271·91-s + 4.84·97-s + 0.665·103-s + 3.26·109-s + 2.09·121-s + 0.198·133-s − 2.72·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(740155.\)
Root analytic conductor: \(5.41583\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 26873856,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.06607290420\)
\(L(\frac12)\) \(\approx\) \(0.06607290420\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 - 57888 p T^{2} + 11606537426 p^{2} T^{4} - 57888 p^{17} T^{6} + p^{32} T^{8} \)
7$D_{4}$ \( ( 1 + 408 T + 1475294 p T^{2} + 408 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 448982340 T^{2} + 140140995061216262 T^{4} - 448982340 p^{16} T^{6} + p^{32} T^{8} \)
13$D_{4}$ \( ( 1 + 11392 T + 1304343618 T^{2} + 11392 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 8867207040 T^{2} + 87318671207303692802 T^{4} - 8867207040 p^{16} T^{6} + p^{32} T^{8} \)
19$D_{4}$ \( ( 1 + 38048 T + 11317055298 T^{2} + 38048 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 297787664580 T^{2} + \)\(34\!\cdots\!22\)\( T^{4} - 297787664580 p^{16} T^{6} + p^{32} T^{8} \)
29$D_4\times C_2$ \( 1 - 1196422681120 T^{2} + \)\(70\!\cdots\!02\)\( T^{4} - 1196422681120 p^{16} T^{6} + p^{32} T^{8} \)
31$D_{4}$ \( ( 1 + 2026808 T + 2265252983058 T^{2} + 2026808 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 3177300 T + 7523056278182 T^{2} + 3177300 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 9778258768384 T^{2} + \)\(12\!\cdots\!46\)\( T^{4} - 9778258768384 p^{16} T^{6} + p^{32} T^{8} \)
43$D_{4}$ \( ( 1 + 9341872 T + 45020015547138 T^{2} + 9341872 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 73842834104260 T^{2} + \)\(23\!\cdots\!82\)\( T^{4} - 73842834104260 p^{16} T^{6} + p^{32} T^{8} \)
53$D_4\times C_2$ \( 1 - 1722356070048 p T^{2} + \)\(87\!\cdots\!26\)\( T^{4} - 1722356070048 p^{17} T^{6} + p^{32} T^{8} \)
59$D_4\times C_2$ \( 1 - 143854028072580 T^{2} + \)\(34\!\cdots\!42\)\( T^{4} - 143854028072580 p^{16} T^{6} + p^{32} T^{8} \)
61$D_{4}$ \( ( 1 + 5875948 T + 136488109567398 T^{2} + 5875948 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 13216208 T - 71593480060542 T^{2} + 13216208 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 485796219665604 T^{2} - \)\(70\!\cdots\!54\)\( T^{4} - 485796219665604 p^{16} T^{6} + p^{32} T^{8} \)
73$D_{4}$ \( ( 1 - 54544160 T + 1922360407753602 T^{2} - 54544160 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 20909320 T - 1319199633536238 T^{2} + 20909320 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 200344099370940 T^{2} + \)\(54\!\cdots\!22\)\( T^{4} + 200344099370940 p^{16} T^{6} + p^{32} T^{8} \)
89$D_4\times C_2$ \( 1 - 9864373835363584 T^{2} + \)\(54\!\cdots\!86\)\( T^{4} - 9864373835363584 p^{16} T^{6} + p^{32} T^{8} \)
97$D_{4}$ \( ( 1 - 214424128 T + 22483442787990018 T^{2} - 214424128 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.873198858013689769665840159618, −8.853877438451441398770393232275, −8.749411793615294523595911205870, −8.259722358542695652031551581942, −7.82516560051463763255171930553, −7.41704121612131487790898894543, −7.33161933164816857347002718764, −6.93398228210923187556612384285, −6.48498194619549093611512253699, −6.39752129260687412255366931571, −6.06792856533871584079278007772, −5.20579064361301137343536508410, −5.12389280379553883982421040044, −4.95080761186791889829647684665, −4.83721325868696982078608726676, −3.82109800166886667358445132618, −3.44947917973613573020204063567, −3.40500130106672237652043563948, −3.25865752348020595167677217982, −2.28100183837452190180446531216, −1.79895528834445289893742006570, −1.79690098208905864570945924298, −1.43724168111169056256351087518, −0.36764775462128692492401683634, −0.06626563435931322772757796014, 0.06626563435931322772757796014, 0.36764775462128692492401683634, 1.43724168111169056256351087518, 1.79690098208905864570945924298, 1.79895528834445289893742006570, 2.28100183837452190180446531216, 3.25865752348020595167677217982, 3.40500130106672237652043563948, 3.44947917973613573020204063567, 3.82109800166886667358445132618, 4.83721325868696982078608726676, 4.95080761186791889829647684665, 5.12389280379553883982421040044, 5.20579064361301137343536508410, 6.06792856533871584079278007772, 6.39752129260687412255366931571, 6.48498194619549093611512253699, 6.93398228210923187556612384285, 7.33161933164816857347002718764, 7.41704121612131487790898894543, 7.82516560051463763255171930553, 8.259722358542695652031551581942, 8.749411793615294523595911205870, 8.853877438451441398770393232275, 8.873198858013689769665840159618

Graph of the $Z$-function along the critical line